Capacitated discrete unit disk cover

Consider a capacitated covering problem as follows: let ... be a customers set of size ... and ... be a service centers set of size ... Assume that a service center can provide service to at most number of customers. Each ... has a preassigned set of customers to which it can provide service. In other words, each customer can get a service from a set of preassigned service centers. The objective of the capacitated covering problem is to choose a minimum cardinality subset such that each customer in ... will get service from at least one service center in ... In this paper, we consider the geometric version of the capacitated covering problem, namely, capacitated discrete unit disk cover problem. In the capacitated discrete unit disk cover problem, the set ... of customers and the set of service centers are two points sets in the Euclidean plane. A service center can provide service to a customer if their Euclidean distance is at most 1. Also, a service center can provide services to at most ... number of customers. We call this problem as -covering problem. For the ...-covering problem, we propose a necessary and sufficient condition for feasible solution of an arbitrary input instance. Next, we prove that the -covering problem is NP-complete for ... We also show that the problem admits a PTAS.(ProQuest: ... denotes formulae omitted.)